Physics - Unit & Dimension Question with Solution | TestHub

The simple pendulum at angular amplitude ${\theta }_0$ is shown in the figure. $T_{\max }=mg+\frac{m{v}^2}{I}\dots (i)$ When, bob of pendulum comes from $A$ to $B$, it covers a vertical distance $h$.$\therefore \cos {\theta }_0=\frac{l-h}{l}$ $\Rightarrow h=I(I-\cos \theta )\dots (ii)$ Also during $A$ to $B$, potential energy of bob converts into kinetic energy $\therefore mgh=\frac{1}{2}m{v}^2\therefore v=\sqrt{2gh}\dots (iii)$ Thus, using Eqs. (i), (ii) and (iii) we obtain $T_{\max }=mg+\frac{2mg}{I}I\left(1-\cos {\theta }_0\right)$ $=mg+2mg\left[1-1+\frac{{\theta }_0{}^2}{2}\right]$ (By using binomial theorem) $=mg\left(1+{\theta }_0^2\right)$